Expected deadlock time in a multiprocessing system - Computer ...

Expected

KEVIN

Deadlock

J.

Time

COMPTON

AND

in a Multiprocessing

CHINYA

System

RAVISHANKAR

Unmermy of Mzchigan, Ann Arbor, Mdzigan

Abstract. We consider multiprocessing systems where processes make independent, Poisson distributed resource requests with mean arrival time 1. We assume that resources are not released. It is shown that the expected deadlock time is never less than 1, no matter how many processes and resources are in the system. Also, the expected number of processes blocked by deadlock time is one-half more than half the number of initially active processes. We obtain expressions for system statistics such as expected deadlock time, expected total processing time, and system efficiency, m terms of Abel sums. We derive asymptotic expressions for these statistics in the case of systems with many processes and the case of systems with a fixed number of processes. In the latter, generahzations of the Ramanujan Q-function arise. We use singularity analysis to obtain asymptotlcs ot coefficients of generalized Q-functions. Categories and Subject Descriptors: D.4, 1 [Operating Systems]; Process Management—deadlocks; multiprocessmg/multtprogrammmg; F.2.2 [Analysis of Algorithms and Problem Complexity]; Nonnumerical Algorithms and Problems—computations on dtscrete structures; G.2. 1 [Discrete Mathematics]; Combinatorics—generatingfunctlons; G.2.2 [Discrete Mathematics]: Graph Theorypfk and circuit problems General

Terms:

Additional analysis

Key

Performance, Words

Theory

and Phrases:

Asymptotic

analysis,

expected

time

analysis,

singularity

1, Introduction Deadlock cessing

detection systems

for example)

and resolution

(see Bic

deadlock

and

is rare

Shaw

is a major [1988]).

issue in the design

Although,

in some

and the cost of resolution

(such as database systems) the likelihood resolution requires an expensive rollback

of multipro-

systems

(Unix,

is low, in many

others

of deadlock may be quite high and and recovery. It would be useful to

know under what circumstances deadlock is likely and (especially when resolution is costly) the expected time for the occurrence of deadlock. This paper presents requests

a model of multiprocessing systems where processes make resource independently and with Poisson distributions of mean 1. We derive

K. J. Compton would like to express his appreciation to the Institut Informatique et en Automatique and l’Ecole Polytechnique for partial this research was done. He would also like to thank the members of prowdmg a supportive research environment and good camaraderie year. Authors’

present address: EECS Department,

University

of Michigan,

National de Recherche en support during the period Projet Algo at INRIA for in the 1991–92 academic

Ann Arbor,

MI 48109-2122.

Permission to make digital/hard copy of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proht or commercial advantage, the copyright notice, the title of the pubhcation, and its date appear, and notice is given that copying is by permission of ACM, Inc. To copy otherwise, to republish, to post on servers, or to redistribute to lists, requires prior specific permission and/or a fee. @ 1995 ACM 0004-5411 /95/0500-0562 $03.50 Journal of the As.o.,dtmn fm COmputmgMachinery,Vd 42, NO 3, Mq 1W5, pp 562-587

Expected

Deadlock

Time

563

o

❑

P,

P2

P3

P4

..

FIG. 1.

P5

A resource

allocation

graph.

P6

PI

•1

P8

Resources

exact

Processes

and asymptotic

deadlock,

expressions

expected

simplifying

total

assumption

be viewed

as upper

Let us describe composed

that

bounds

time,

resources a little

of entities:

statistics

such as expected

and system

are never

or bounds

our model

of two types

for system

processing

efficiency.

released.

for an extreme

more

precisely.

processes

Thus,

time

We make our results

case of system

may

behavior.

A multiprocessing

and resources.

to the

system is

Processes

are the

active entities of the system. They can change the system state by requesting new resources or releasing resources allocated to them. Resources are serially reusable: they may be reallocated once they are released. Examples of such resources are hardware units such as memory pages or printers, and software resources such as database locks. We do not examine systems with consumable resources process

such as messages, requests

only

signals,

one resource

A system state is represented graph

whose

is bipartite;

vertices edges

are

free,

p releases an edge

released. requests inactive appears

from

We also assume

that

each

been

and resources resources allocated

graph.

This is a directed

in the system.

to processes

The graph

or processes

to a process.

If a process

to p

r, an edge is inserted in the resource allocation graph that r has been allocated to p and is no longer free.

r the edge is erased.

is drawn

In this

directed

1). if it has not

requests a free resource from r to p to indicate

data.

by a resource allocation

are the processes

resources (see Figure A resource is free

When

and input at a time.

from

If p requests

p to r, indicating

case, p becomes

inactiue

that

or blocked

a resource

r that

p is waiting

for

and can make

is not r to be

no more

until r is released. Thus, an active process has out-degree O and an process has out-degree 1. Deadlock occurs when a directed cycle in the resource allocation graph. Since all the processes on the

directed cycle are blocked, the resources on the cycle can never be released. They become useless until the deadlock is detected and resolved. For example, if process p ~ requests resource r4 in Figure 1, deadlock occurs because a cycle

K. J, COMPTON

564

AND

C. RAVISHANKAR

p7 + r4 + p2 + r2 + p3 -+ T5 + p7 results when the edge p~ ~ rd (indicated by the dotted arrow) is added. A more detailed description of the model is given in the following section. Lynch [1994], in enumerating the number of cycle-free resource allocation graphs for given numbers of resources necessary and sufficient conditions for resource

allocation

graph.

(i)

Every

vertex

(ii)

Every

resource

To see this,

and processes, used the following a bipartite directed graph to be a

has out-degree with

observe

at most

non-zero

that

one.

in-degree

a resource

has out-degree

acquires

exactly

an out-edge

one,

the first

time

it is

requested; thereafter, it acquires only in-edges. A process, on the other hand, acquires in-edges until it acquires its first out-edge; thereafter, it is inactive. (For more general models of deadlock and associated references, see Bic and Shaw [1988].) From his enumeration, Lynch model where resource allocation edges are equally to problems

likely.

such

Flajolet

[1990]),

[1988]

and Flajolet

Our

as linear

[1994] derived some system statistics for a graphs with the same number of directed

model probe

computation

is quite hashing

of random

mapping

[1990a]),

analysis

Yao [1976] and Knuth and Schonhage Knuth [1985]). These problems are all

[1978]), analyzed

called Abel’s The

because identity). outline

and Odlyzko

different.

many

of them

of the paper

and derive recurrence note that all of these terms

of Abel

can be evaluated is as follows:

from

a result

any state with

[1973]

statistics

and optimum in terms of explicitly

In Section

related

Vitter

and et al.

algorithms

(see

caching (see Abel sums (so

by generalizations

2, we describe

our

of model

statistics. In Section 3, we form and give solutions in

we call the Half j active

closely and

(see Flajolet

of union-find

relations for various system recurrences have a common

sums. We prove

says that beginning

It is more

(see Knuth

and Half

processes,

Theorem.

the expected

It

number

of these processes blocked by deadlock time is (j + 1)/2, one-half more than half the number of initially active processes. In Section 4, we develop a general theory for the evaluation of Abel sums. This is applied to the recurrence solutions of Section 3 to give expressions and inequalities for system statistics. In Section 5, we give asymptotic expressions for these expressions in the case where

systems

where systems the Q-function

have many

processes.

have a fixed number arise, This function

In Section

6, we do the same for the case

of processes. Here, functions was studied by Cauchy [1826],

generalizing and later by

Ramanujan [1912]. Ramanujan actually denoted it 6’ (as he did several other functions). Knuth was the one to name it Q in his first expected time analysis of an algorithm [Knuth 1968, pages 113–1 18] (see also Flajolet, et al. [1992]). We use singular expansions to obtain the asymptotic of coefficients of functions of this type. We observe several notational conventions. The expression [z” ]s(z) denotes the rzth coefficient of the generating function s(z). The expression nn denotes the falling

factorial

rz(n – 1) .”. (n – m + 1). Stacked

numbers

in braces

/:;, )

denote Stirling numbers of the second kind (or in the terminology of Graham et al. [1989] the subset Stirling numbers). The expression j!! denotes the double factorial function 1.3 “ 5 “”” (2j – 1). E(M is the expectation of the random

Expected

Deadlock

Time

variable

X. We write

another

way of writing

2. Recurrence In this

~ < g if ~(n)/g(n) ~(n)

approaches

O as n ~

~; this

is just

= o(g(n)).

Equations

section,

recurrence

565

we give

relations

a more

for system

detailed

description

of our

model

and derive

statistics.

Let m be the number of resources and n be the number of processes in a multiprocessing system. Suppose that the system has just entered a state where there are i free resources and j active processes. Let ~, ~ be a random variable representing the time to deadlock. ~, ~ depends on m as well as i and j, but for notational

convenience

we suppress

m. We shall

see that

~, ~ does not depend

at all on n. Consider active

the

process,

able with

arrival

mean

exponentially

time

this value

1. For j active distributed

calculation

shows that

mean

That

j-l,

for

the

would

next

request.

processes,

random

variables

express

the

with

mean

distributed

arrival

were

just

distributed

it is the minimum

this is an exponentially

is, we may

If there

be an exponentially

time

a single

random

vari-

of j independent, 1. A

straightforward

random for

the

variable

next

with

request

as

j-lXl,,, where X,:~ is an exponentially distributed random variable with mean 1. Thus, after a time interval of length j- lX,,,, a process p chooses a resource r. Now r may be one of the i free resources or one of the m – i allocated resources. where serted free

The

P[A,,,

event

that

r is free

is represented

by a random

variable

~,,,

= 1] = 1 – F’[xl,,l = 0] = i/m. When A,,l is 1, an edge is mr to p and the system enters a new state in which there are i – 1

from resources

edge inserted

and from

j active

processes,

p to r would

result

When

zl,,~ is O, we check

in deadlock.

to see if an

We do this by constructing

a directed path from r as follows: Follow the unique out-edge from r to a process. If this process has out-degree one, again follow the unique out-edge. Continue in this fashion. The resource allocation graph contains no directed cycle yet, so this path terminates. In fact, it terminates at a process, since resources with non-zero in-degree have out-degree one. If the process in which this path terminates is p, an edge from p to r would complete a directed cycle and deadlock processes, not occur represented (j

– I)/j.

results.

we do not

If the path have

is (j – 1)/j. by a random Putting

terminates

deadlock.

Thus,

at any of the the probability

other that

j – 1 active deadlock

does

The event that deadlock does not occur may be variable B,, ~ where P[ B,, ~ = 1] = 1 – P[B,, ~ = 01 =

this all together,

we have (1)

where the random variables Al, ~ and B,, ~ are independent of each other and of the random variables X,,,, ~_ ~ ~, and ~, ~_,. This model is counterintuit’ive in one respect. Suppose that process p requests a resource r and at some later time p requests r again. This is not precluded in our description of the model. The system enters deadlock at this point because a 2-cycle appears in the resource allocation gralph. If we do not allow the possibility of a process requesting a resource already allocated to it, then the system may never reach deadlock. It may instead reach a state in which all resources have been allocated to a particular process and all other processes are blocked. If we wanted to forbid the possibility of a process

566

K. J. COMPTON

requesting

a resource

already

allocated

about what we mean by expected for considering the former model

to it, we would

AND

C. RAVISHANKAR

have to be more

careful

deadlock time. Of course, the main reason is that it is analytically more tractable.

In the case in which there is a single famous birthday problem (see Krmth

process, [1968]).

our model is an instance of the This problem asks how many

people one must choose at random to find a pair with the same birthday. is equivalent to labeling 365 resources with the days of the year (we ignore days) and computing occurs

only

number

a resource

of people

Closely problem

the expected

when

one must

related

deadlock

choose

to the problem

of finding

the total

time,

is repeated.

In

which,

for a year with

of finding

processing

with

general,

just

one process,

Tw, ~ is the

expected

m days.

the expected deadlock time is the This requires P,,, before deadlock.

time

only a minor modification if j processes are active

of the argument used to derive eq. (1). Observe for a time interval given by j- IX,,,, then the

processing

interval

time

over that

=Xl,,

~,, Now using

if we let linearity

is Xl,,.

+A[,,~_l,

~,1 = E(~,,)

that, total

Hence,

J + (1 –At,

J) Bz, jP,,, _l.

P,,l = E(P,, I ), from

and

of expectation

This leap

and independence,

then

(2)

(1) and

(2) we have,

multiplying

by j,

(3)

(4) for ~ >0. A system measures in which total

statistic

system

j processes

processing

processes We will

that

will

efficiency.

be especially Note

that

is total

a time

interval

are active

for

to this

quantity

time

important

jT1,,

is the

is ~ ~ = P,, ,/j~, processing

time

of T,,,. The

expected

ratio

fraction

~ which

in a system of actual

of time

that

are active. also be interested

in the variances

of

~,,

and

P,,,. Therefore,

we

wish to compute the expectations ~ ~ = E(Tl~J ) and Q,,, = E(P,~, ). Squaring both sides of (1) and using the identities A,, j(l – Al,,) = 0, A~,, = A,,,, (1 – A,,,)z = 1 –A,,,, B~l = B, ~, we have

+2j-]X,,, Now take expectations

A,,, ~_l,,

+ 2j-lx,,

and use eq. (3). Note

J(l that

–A,,,~B, E(X~,)

,~,l-I.

= 2. Then

multiply

by

j to obtain

jVJ Performing

= 2T,1

the same series

i + ~j~-~,l

+

of computations

l–~ (

(j–l)~,,.l.

(5)

“1 beginning

with

(2), we have

(6)

Expected

Deadlock

Time

567

SJmlml

Descrlptlon

s LJ

z]

Expected

IT, ,1

1

(3)

j Pi.:

J

(4)

‘j U,,j

2T, ~

(s)

jQ,,,

2JP,,l

(6)

JR, ]

t~/m

(7)

x,,

Equdt ]on

Deadlock T]me e,

Expected Total Processing Time

uJ]

Expe( ted Square

FIG. 2.

Summary

of recurrences.

of Deadlock Time E\pected

Q, J

Square of

Tot al Processing Time R,,

Expected Number of Resources Allocated

The last statistic we consider resources. This is also important deadlock

time

requesting expected occurs.

is fairly

long,

resources number

This

measures how well the system for system design. Even if the

system performance

are usually

blocked.

if processes

therefore,

in the

of resources

a resource,

.. jR,, j = xm

the probability

i + ;jRl_l,,

that

+ (

The

be poor

are interested,

R, , that will be allocated by the time deadlock is easy to determine from our model. Notice that whenever a

process requests i/m. Thus,

3. Solving

still might

We

allocates expected

the General

eqs. (3)–(7)

the request

1 – ~ (j m“)

will

be granted

– l) R,,J_l.

is

(7)

Recurrence

have the form St,, = X,,j

+

+( “)

h,_l>J m

1 – ~ m

st,, _,,

(8)

where

S,, ~ = Xl, ~ = O for all i. (Note that there is no need to specify the values of so , since SO,~ = Xo, ~ ~ So, ~– 1.) We summarize the results of the previous section, with respect to this general form, in Figure 2. Recurrence (8) is linear in Xi, j; that is, if X,,, = aX~,, + bX~, for all i and j, then S,,, = aS~,, + bS,,, where $,, and S: ~ are solutions of the obtained from (8) by replacing X,,, with X;, ~ and X:,, respectively. To solve (8), we might and solve the resulting to this equation

function. .

St(z)

easily

Instead,

generating

equation.

for s,(z)

=x,(z)

x,(z)

+ q(z) m

gives an expression

+ in terms

x,(z)

= m–(m–i)z

=

j >1

for coefficients

of generating ixl,lz~. 1=1

to obtain

( “) 1 – ~ m

of s,_ Jz)

2$(Z). and xl(z): st_l(z).

+ m–(m–i)z

Z,,, S,,, y’z] the solution

i

m s,(z)

solution

sequences

and

~ St,,.d ]=1

function

Unfortunately,

to a closed-form we form

eq. (8) by z~ and sum over the range s,(z)

Solving

=

a bivarlate

differential

does not lead

of the generating

Multiply

try to form partial

equations

functions

568

K. J. COMPTON

Substitute for s,_ ,(z) on the right sl _ Z( z), and so on. We have

s,(z)

side of this

=

AND

expression,

C. RAVISHANKAR

then

substitute

at, k(z)x~(z).

~

for

(9)

k=O

where n’zi(i a,, k(z)

1)...(k + 1)

-

= (m

-

(m

-i)z)(m

+ 1)2) ”””(WZ – (m -k)z)

– (n? -i

It is instructive to use (9] and (10) to compute expectation in the birthday problem for a year with i > () and j > 0. Then, T~ ~ is the linear coefficient sum of the constant

We may regard jan Q-function. a family

of the Q-function In a later section,

– 2)

+

. . +(~

-

l)(m -2)...1 m—1 m

of m, The function T~, ~ – 1 is the Ramanucoefficient of s~(z) and thus is just the first

given

used by Knuth

T~, ~, which, recall, is the m days. Take Xl,, = 1 for of s~( z) and hence is the

m’

this as a function T~, ~ is the linear

of functions

techniques

(lo)

of am, ~. Thus,

(7?’2– l)(nl

m–l ———— + m

Tm.1 =1+1+

of

coefficients

“

by the

and others

successive

to obtain

coefficients

of

the asymptotic

s~(z).

The

of coefficients

do not seem to extend easily to other functions in the family. we will develop other asymptotic methods to deal with these

functions. Let tion,

us return

to our analysis

of (9) and (10). By partial

fraction

decomposi-

Thus,

[2’1%,(2)

=

i

(-l)

’-’(i

_ ,,/_

~,k)(l

-

;)[-’+’

l=k

and consequently

s,,,

=

~ ‘il([z’laL, k=or=o

k(z))

([z’-’

lxk(z)) L–k+r

_ —

~’~’ k= Or=

=

=

}.(;)

i%+

~=1

&l)

’-k(i_l,

;_~,

~

l–~ )(

Ol=k

:O(;](;

-

i

1=1

(f)>o(~j(;

l)

’-k:~(l

-

-

1)[-’2J

m

‘k]-r

)

;)[-’+’%

-

;]

’-’’’x,,,_r.

,11,

Expected This

Deadlock

appears

considerable

a little

cumbersome,

then

be evaluated introduce

but

for

In particular,

many

choices

of

X,,J

if X, ~ can be written

there

is a

as a product

we can write

1{ = 1 – l/m.

The

separately.

sums

Z~. ~(~)y,(-ll)i-’

Results

below

and

follow

Z;j~ZJ.,l~

in this way.

can then

It is convenient

to

the notation

c,(P, Note

569

simplification.

X,,, = ~Z,

where

Time

that

explain

!l)

Ji)(:)[-p(l$+’

=

z

+ 1,q + 1) = Ci(p

c,(p

(13)

~

+ 1,q) – c,(p,

how Ci(p, q) may be computed

in closed

q).

form

In the

next

section,

for some values

we

of p and

q. 3.1.

THEOREM

expected deadlock (i) (ii) (iii)

We have the following

total processing time.

time,

and

expressions

expected

for

number

expected deadlock

of resources

time,

allocated

at

~,, = 1 + (Ct(l, O) – Cl(l, j))/j. 1)/2 + C,(l, O) - (C,(2, 1) - C,(2, ~ + I))/j. P, R:,, = c,(l, O) – (c,(2, 1) – c,(2, j + 1))/”. j

=

(~

+

For

PROOF.

Binomial

(i),

Theorem

take

(ii)

(8)

so that

of geometric

eq.

series

1( = = 1(1 _ ~)~-1

(;)k(-~l)’-k

k=O

]–1

1, – q+l (15)

~(j-r)I[=&~=o which Parts

are derived (ii)

by standard

and (iii)

1 ❑

techniques.

of the theorem

(1 - 1{)2 ‘

already

provide

useful

information

about

system behavior. The expected total processing time Pi, ~ is also the expected number of requests since the request arrival time for each process is 1. The number of resources difference between this value and R,, ~, the expected allocated, is the expected number see that this quantity is always (j

of processes blocked by deadlock time. We + 1)/2, one half more than half the number

of initially active processes. system performance.

we have the following

THEOREM

3.2

(THE

with j active processes, time is (j + 1)/2.

HALF

Thus,

AND

HALF

the expected number

THEOREM).

surprising

Beginning

result

from

of these processes blocked

about

any state

by deadlock

570

K. J. COMPTON

AND

C. RAVISHANKAR

There is an easier way to prove the Half and Half Theorem. The theorem is clearly true when j = 1. Now suppose the system is in a state where the number

j of active

Theorem number

holds for states with j – 1 processes. The system may change states a of times before it changes from a state with j active processes to a

state with of active state l/j;

processes

j – 1 active processes

change.

is more

processes.

decreases

The

in this event, just

Let

critical.

probability

than

one and assume

us call state A system

of deadlock

changes

where

can deadlock

at the

one of the original

the Half

first

processes

the number

only

critical

and Half

at a critical

state

is blocked.

change

is

The probability,

then, that deadlock does not occur at the first critical state change is (j – 1)/j; in this event, the system enters a state with j – 1 active processes and the expected

number

of remaining

processes

that will

be blocked

by deadlock

times

is j/2, by the induction hypothesis. Thus, the expected number of processes blocked by deadlock time is (l/j) + (1 + j/2)(j – 1)/j = (j + 1)/2. Using these ideas, we can determine how many processes must be blocked for the probability of deadlock to exceed deadlock after k critical state changes is 1 ~+j–l

j–l

l”””

3.3.

Before (11). free.

If we allow

concluding

Consider Changing

““”

result

a system with j initially

are blocked,

this

section,

k

1

j–k+2j–k+l=~

so we set k = tij. We have the following

until 8j of those processes is 8.

of reaching

+ j_lj_’2

j–k+l j_

probability

1

j

j_lj–’2

+——j

THEOREM

The

1 j–lj–2 —+———

j

J

8>0.

let

the probability

us explore

actiue processes that deadlock

another

the case where i = m, that is, where the order of summation we have

to operate

has occurred

consequence

all resources

of eq.

are initially

The last equation follows from formula (6.19) of Graham et al. [1989]. Although this is a more pleasing form than eq. (12) (and shows, in particular, that of the quantities X~, ,), it is not as useful. s m.] is a positive linear combination

4. Evaluation

of Abel

Sums

Abel’s identity is expressed in several different one of them (see Riordan [1968, page 18]): 1 xx [=0

j (1

(x+l)l-

l(y+i–l)’-l=

forms

in the literature.

(.x+y+

i)’.

Here

is

Expected

Deadlock

It follows

Time

immediately

571 that

1.

(x+y+i)’–(y+

X()( ;

x+l)l-l(y+i–

x

1=1 Taking

the limit

as x approaches ‘~(~)l’-’(y 1=0

If we put

O we have +i

-1)

’-’

=i(y

+i)’-’.

y = m – i and

needed to evaluate [1968, pages 18-23] give some

divide by m’-l, we have the expression in Theorem 3.l(i), considers several generalizations

of the other

some

typographical

good

asymptotic

i)’

l)’-’=

one of the cases of (13) viz., CL(l, O) = i. Riordan of Abel’s identity. These

cases c,( p, q) by the same argument.

errors results,

in Riordan’s we must

formulas.)

develop

(N.B.

However,

a more

general

There

in order theory

are

to get

for

expres-

that

C,(p,

sions of the form

and

then

evaluate

c,(p,

q, Y) is a convolution.

fp(~) =

q) = Cl(p,

q, m – i)\m’-P+~.

First

note

Let

,:1:=’

and

gq(z)

m (y + k) ’’+qz, x k! k=o

=

“

(If p = O, then it will be convenient to begin the summation for ~p(z) at k = O rather than k = 1 so that the constant coefficient is 1.) Then CL( p, q, y) = [z’/i!]fp(z)gq(z). The function ~l(z) = ~(z) is well enumeration. A standard textbook application of Formula is to show that jlz) is the solution of

known in combinatorial the Lagrange Inversion the functional equation

~(z) = zef(z) (see Wilf [19901). If we differentiate both sides of this substitute ef(z) = ~(z)/z, and solve for ~’(z) we have zfl(z) = jlz)/(1

equation, – ~(z)).

Thus, f,(z) from

which

= ~

Zf(x) y

it follows f,(z)

dx = J : (1 –f(x))f’(x)dx

=f(z)

– ;f(z)~,

that f,(x) yal

“

~

— .

z J-(

1 – :f(x)

o

=f(.z)

– ;f(z)’

If we use the Lagrange Inversion e ‘f(z) and then substitute f(z)/z for

)

(1 – f(x))

fl(x)

+ ;f(z)s. Theorem to compute ef(z), we have k–1

f(z)’

=Y

:

h=l

ah

‘y‘k:)

Zy+k,

the

coefficients

of

572

K. J. COMPTON AND

Differentiate,

multiply

by

z,

and

replace

z~’(z)

with

_

f(z)

y

–

— (1 z

C. RAVISHANKAR

~(z)/(1

– ~(z)).

We

obtain

m (y+k)’zk go(z)= It follows

~ L=o

~, “

1 1 –f(z)

“

that

fl(z)go(z)

‘+’

f(z) — () z

= z

1 1 –f(z) k– 1

=

5

‘(y

y

Zk>

k=l

and

so

C,(l,

~,(z) go(z)

We p >0.

O,y)

= i(y

+ i)[-l.

A

similar

evaluation

of

~2(z)go(z)

and

shows that c,(2,0,

y)

= i(y

+ 0-1

–

1 #(y

+ i)’-z,

c1(3,0,

y)

= i(y

+ 0-1

–

;iqy

+ ~) ‘-2

can use the method We can show easily

above to compute by induction that

f,(z)

=

i

+ :ij(y

values

of

+ ~)’-s. C,(p,

O, y) whenever

D,,kf(z)’.

k=l

where

for

all positive

p,

DP, I = 1 and

DP+ ~,k+ ~ = (DP, k+l

– DP, ~)\k.

These

coefficients are the “differences of reciprocals of unity” appearing on page 248 of David et al. [1966]. (This source was located with the help of Sloane [1973]; cf. sequences

2049 and 2305.) C,(p,

It follows

o,y)

=

f

that Dp,,iqy

+ i)’-k.

k=l

We can now compute values are as follows. c,(l,

the values

q) when

p > q >0.

The

first

O) = i, i(i

c,(2,0)

of c,(p,

– 1)

= im – 2’ 3i(i

c,(3,0)

i(i

– 1)

= imz –

– l)(i

4

6’ i(i

c,(2,1)

= c,(2,0)

– 2)

m+

– cz(l,

+ 1)

O) = im – 2’ (3i

c,(3,1)

= c,(3,0)

– c,(2,0)

+ l)i

= imz –

~

4 = c,(3,1)

– c,(2,1)

= imz – 4

(i + l)i(i

– 1)

6’ (i + 2)(i

(3i + 5)i c,(3,2)

+

m+ 6“

+ l)i

few

Expected

Deadlock

Time

573

In general, P Ct(p,

and values

of Cl( p, q) for

Substituting ing theorem

o)

=

~ k=l

qzi%rz~-~

1 < q < p are obtained

these values for giving expressions

by differencing.

Ci( p, q) into Theorem 3.1, we have the followfor various system statistics in terms of Abel

sums. THEOREM

4,1.

expected number (i) (ii) (iii)

~,,

The expected deadlock of resources

= 1 + i/j

P,,, = (j

-

+ 1)/2

+ i(i

+ i(i

+ 1)/2j

eq. (13), we see immediately

COROLLARY

+ 1)/2j

that

total processing

time,

and

time are as follows:

+ c,(2, j + 1)/j.

c,(1, j)

important

The expected

4.2.

upected

+ CL(2, j + I)/j.

c,(2, 1), so we have the following

and expected number

time,

at deadlock

j)/j.

+ i – ire/j

Ri,J = i – ire/j

From

c,(l,

allocated

deadlock

of resources

s c,(1, O) and

c,(2, j + 1)
i\(3(m 1

point

1 + l/(m

– l);

the

– 1)) so for all z on r

3nl 2,

and

f2(z)f-,

-l(z)

=

(j

+

l)!!

– l)(2j

&

–

[ 144j4

+ 1344j3

)

12j2 + 72j + 358-2, 48(2j + 1)

– 216j2

– 144j

– 47

+ 2304(2j+

for j >1.

By the Binomial TV2 r(a/2)

Theorem

~m+(a–

2(”-lJ/2 9a~ – 60a3

8-2]+

1

+. . . .

and Stirling’s

}

formula

[z~/rn

3a2–6a+2 (

1 + &

-1

l)(2j–1)

1)/2

1 +

+ 120a2

+

24m – 72a

1

+ 4 +o~.

1152mz

+ .+O

– 3)

( m

))

!]8-”

is

580

K. J. COMPTON

Substituting

this into

the expressions

above

(2m)l/2 c~(l,

l)

=m

–

~

and simplifying,

C. RAVISHANKAR

we have

(2*)1/~

1 + — – 3

nil/2

AND

~l–l

/’2

24

4 –’ + O(n-slz) + 135 “ 21\zr(j C,,,(l, j)

+ 1/2)

~l,z

+

2j

– 1

= nz – 3

I’(j)

21/~r(j

+ 1/2)

djz

5m-1,2

– 6j +

361’(j)

2j–1

2j2–4j–6 135 c.(2,

1 j + 1) = ~m2

m

-1 + o(n-3/2)

2j+l –

—

2m 21\zr(j

+ 1/2)

2j

+ 1

~,z

j2

+ r(j) The

asymptotic

formula

for

Cn(l,

— 3 1) agrees

m

with

+ o(n-”2)

–3

eq. (25)

[1968]. Our approach seems somewhat simpler analysis of the incomplete gamma function. From the main result of this section.

in 1.2.11.3

since it Theorem

of Knuth

does not require 4.1, we now have

THEOREM 6.1. Fix j. For systems beginning from a state with j processes and all resources free, we haz,)e the following asymptotic formulas for expected deadlock time, expected total processing time, expected number of resources allocated at

deadlock

time,

and system efficiency.

(2=)1/2 T m,l

2 + — + 3

ml/2

=

2 2’12r(j T

m>] =

(2m)”2 24

+

0(n-31~),

135 m

+ j + 1

+ 1)

2’/2(4j2

‘1

‘n

+ l/2)m1,q

r(j

4

–1/2

-

3j

– 6j + 5)1’(j

– 1/2)

+ 72r(j

_1,2 m

+ 1)

2jz–4j–6 +

135j

23/211j P

m,] =

+ 3/2)

3r(j

-1 +

311j

+ 3/2)

=:

+

~

+ O(m-1/2), 6 mll~

+ 1)

+

when

m1,z + j + 3

1 F m.]

o(n-3/2),

+ 1)

23\217(j R=m,J

‘z

o(l’wl/~).

– ;

“

+ @m-1/2),

j >2

Expected

Deadlock

Thus,

we find

processing than

that

time,

constant 2/3.

mined

Time

581

for

and

multiple

of nzl\2.

This

seems very

the order

a fixed

number

of

System

number

allocated

efficiency

promising.

of the variances

of processes,

resources

always

deadlock

and total

not

total to

a value

we have

time

time,

asymptotic

approaches

Unfortunately,

for

deadlock are

a

greater

yet deter-

processing

time

and we have no idea of the distribution. 7. Conclusions Our results suggest detection at regular

that it may be worthwhile intervals in some systems.

for this is the absolute 4.2(i)). Also, the high processes

(Theorem

lower values

5.l(iii))

bound for for system and

systems

(Theorem 6.1) are encouraging. If we knew the distribution of deadlock

to investigate doing The most compelling

expected efficiency with time

deadlock evidence

deadlock time (Corollary for systems with many

a fixed

number

~ ~ we could

of processes

determine,

for

a

given 8>0, the length of time the system must operate for the probability of deadlock to be i3. We could then compare the cost of continuously updating to do deadlock detection versus the expected cost of resolution and rollback when doing deadlock detection at regular intervals. Even though we do not know the distribution,

Theorem

3.3 provides

a similar,

but

less satisfactory,

approach.

We can count blocked processes. This requires some communication processes and may therefore cost more than waiting for a fixed time but it is still

likely

to cost less than

deadlock

prevention.

What do we have to do to determine the distribution i or j increase? A standard approach for determining random eq. (l), T: l]

This

between interval,

of deadlock time ~ , as the distribution of a limit

variable is to find its characteristic function (see Feller [1971]). ~rom we have the following recurrence for the characteristic function ~ , of

may

generating

lead

to

function

There

are other

a solution, methods important

but

it

does

not

seem

to work

well

with

the

used here. questions

First, although practice.

the absolute lower bound we theoretically optimal, certainly The most obvious reason for

sources

in multiprocessing

systems

still

to be answered.

derived for expected does not come near this is that processes

so deadlock

time

is usually

deadlock time, what occurs in do release remuch

greater

than in the model here. We need to develop a more realistic model, but this will not be easy. It will require adding a queuing-theoretic dimension to the combinatorial problem of determining when cycles emerge in certain kinds of random

bipartite

Second, We would

graphs.

we need to deal with the problem of 2-cycles mentioned expect processes to remember the resources allocated

in Section 2. to them and

to avoid making duplicate requests. As we noted, if we prohibit 2-cycles, then it is no longer clear what is meant by expected deadlock time. It is also not clear how to modify the model. Should we proceed as before with a process randomly choosing a resource but have the system to continue in case a 2-cycle arises? Or should we require a process to choose randomly among resources

582

K. J. COMPTON

AND

C. RAVISHANKAR

not allocated to it? In either case, the model becomes much more complicated. We must keep track not only of the number of active processes, but also of the number of resources allocated to each active process. In addition to the cases in Sections 4 and 5, we would like to have asymptotic expressions

for system

4 if conditions i = o(j).

like

We would

statistics

when

m = o(j)

and

i = Kj. We would

like to know

m log log m = o(j)

like to solve the general

in Section

can be replaced

case in Section

with

5, not just the case

i = m.

Finally, for

we would

fixed

j.

There

like to know are

some

the variances unpublished

and distributions results

on

of ~ , and P,,,

certain

cases

of

this

problem. In the case j = 1, variance and distribution can be determined. This problem should not be confused with one related to the problem of determining the variance and distribution for the traditional birthday problem. Even though

the mean

pair with

of Tn , is the expected

the same birthday

models

the “street

corner

and asks passers-by

corner

for

number

of people

m day years, variance

birthday their

problem”: birthdays

A person until

needed

behaves stands

a repetition

times of passers-by are exponentially distributed random malized mean 1). In the traditional birthday problem (the

to find

differently.

a

Tm, ~

on a busy street is found.

Arrival

variables (with nor“classroom birthday

problem”), a teacher in a classroom aski students their birthdays until a repetition is found. Arrival times are precisely 1. Clearly, the variance for the street corner birthday problem is greater than for the classroom birthday problem. Donald Knuth (personal communication) has informed us that using techniques asymptotic

in Knuth [1985] one may show that variance in to (4 – m )m /2. Guy Louchard of the University Libre

(personal communication) has informed us that, for street corner birthday problems, the random variable suitably variable. tion)

normalized, Alfredo

converges Viola

has some preliminary

ACKNOWLEDGMENTS.

in distribution

of the

We

University

results

on the variance

would

like

to

both the classroom and for a repeated birthday,

to a Rayleigh of Waterloo

thank

this case is de Bruxelles

distributed

(personal

of ~,,,,

Philippe

for fixed

Flajolet

random

communicaj.

for

many

useful suggestions and discussions. We would also like to thank Yuri Gurevich and Andreas Blass for making a fundamental observation that led to the development of the model in eq. (l). Both authors would like to thank the creators of Maple system. out or checked in Maple.

Most

of the computations

in this paper

were

carried

REFERENCES BIC, L., AND SHAW, A. C. 1988. The Logical Design of Operating System&. Prentice-Hall, En@ewood Cliffs, N.J. CAUCHY, A. L. 1826. EXWCWS de Ma fhematzgue. Chez de Bure Freres, Paris, France. DAVID, F. N., KENDALL. M. G., AND BARTON, D. E. 1966. ,Syrnnletnc Funcfzon and Alhed Tab[e~. Cambridge Unwersity Press, Cambridge. England. FELLER, W. 1971. lntroductum to Probabd@ Theory and IfA Applicatiotu, vol. 2, Wiley, New York. FLA.JOI ET, P., GARDY, D., AND THIMONIER, L. 1988. Probabilistic languages and random allocaLanguages and Programmmg, T. Leplsto and A. Salomaa, eds. Lecture tions. In Aufornakz, Notes m Computer Science, vol. 317. Springer-Verlag, pp. 239-253. FLAIOLE r, P., GRABNER, P. J., KIRSCHENHOPER, P., AND PRODINGER, H. 1992. On Ramanujan’s Q-function. Rapport dc Rechercbe 1760. Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France.

Expected

Deadlock

Time

583

FLAJOLET, P., AND ODLYZKO, A. M. 1990a.

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or C~ptol-

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Disc. Math. 3, 216-240. GRAHAM, R. L., KNUTH, D. E., AND PATASHNIK, O. 1989. Concrete Mathernadcs. Addison-Wesley, Reading, Mass. HENRICI, P. 1977. Applied and Computational Corfzplex Analyws, vol. 2. Wiley, New York. KNUTH, D. E. 1968. The Art of Computer Programming, vol. 1: Fundamental Algorithm~, 2nd ed. Addison-Wesley, Reading, Mass. KNUTH, D. E. 1973. The Art of Computer Programmmg, vol. 3: Sorting and Searching. AddisonWesley, Reading, Mass. KNUTH, D. E. 1985. An analysis of optimum caching. J. Algorithms 6, 181-199. KNUTH, D. E., AND SCHONHAGE, A. 1978. The expected linearity of a simple equivalence algorithm. Theoret. Comput. Sci. 6, 281–315. LYNCH, J. 1994. Random resource allocation graphs and the probability of deadlock. SL4M 1 Disc. Math. 7, 458-473. MARKUSHEVICH, A. I. 1977. Theory of Functions of a Complex Variable. Chelsea, New York. Revised. (English edition translated and edited by Richard A. Silverman.) RAMANUJAN, S. 1912. Question 294. J. Indian Math. Sot. 4, 151-152. RIORDAN, J., 1968. Combinatorial Identities. Wiley, New York. SLOANE, N. J. A. 1973. Handbook of Integer Sequences. Academic Press, San Diego, Calif. WTTER, J. S., AND FLAJOLET, P. 1990. Analysis of algorithms and data structures. In Handbook of Theoretical Computer Science, vol. A Algorithms and Complexi~, J. van Leeuwen, ed. North Holland, Amsterdam, The Netherlands, pp. 431-524. Academic Press, San Diego, Calif. WILF, H. S. 1990. generatingfanctionology. YAO, A. C.-C. 1976. On the average behavior of set merging algorithms. In Proceedings of the 8th ACM Symposium on Theoiy of Computing (Hershey, Pa., May 3–5). ACM, New York, pp. 19~_195.

RECEIVED JANUARY 1993; REVISED SEPTEMBER 1994; ACCEPTED NOVEMBER 1994 JournalOfthe As~ocmaonfor COmp~tmgM~chmeq,VOI 42,NO 3, Mq 1995